It is well-known that a local ring is Gorenstein if and only if it is Cohen-Macaulay and its Cohen-Macaulay type is equal to one; in a sense, almost Gorenstein rings are meant to extend Gorenstein rings to arbitrary type. The properties of this class of rings have been studied by several authors in the last two decades (cf. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]} for example); in particular, curious patterns started to appear in works dealing with the Cohen-Macaulay type of almost Gorenstein rings. In the original context of one-dimensional analytically unramified rings, including local rings associated to monomial curves, it is well-known that if the embedding dimension is at most 3 then the Cohen-Macaulay type is either 1 (and thus the ring is Gorenstein) or 2 (cf. [8]}), while if the embedding dimension is at least 4, there is no upper bound for the Cohen-Macaulay type (cf. [8]}). On the other hand, if we restrict ourselves to almost Gorenstein rings of embedding dimension 4, Numata (cf. [10]}) asked if the Cohen-Macaulay type is at most 3, which turned out to be true (cf. [11]}). Families of almost Gorenstein rings with arbitrarily large Cohen-Macaulay type \(t\)  are present in the literature (cf. [12]}, [4]}); however, for these families of rings, the embedding dimension \(e\)  is at least \(\frac{t}{2}\)  (actually, in this context there is no example in the literature of an almost Gorenstein ring satisfying \(t > 2e\) ). Therefore, contrary to the general context, for almost Gorenstein rings and curves the following question (cf. , [10]}, [11]}) naturally arises.
